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“The name on your back, it’s a big bull’s-eye,’’ says Bo Greenwell, 21, who bats third and patrols left field for the Lake County Captains, a low Single A affiliate of the Cleveland Indians. “Every time I strike out every team seems to cheer a little bit louder. When I make a mistake, people seem to laugh a little bit harder. You are definitely under a microscope.’’

In early May, Greenwell was hitting over .400 and leading the Midwest League in both batting average and RBIs. His father, Red Sox Hall of Famer Mike Greenwell, and mother, Tracy, a nurse, made a surprise visit. Bo had tonsillitis and a 103-degree fever and was sent home, but begged his way back into the lineup the next day and went 0 for 4.

His dad was smiling. “I’ve never been one of those parents who scream and yell at the ballpark,’’ says Mike. “I just laugh and tell him if you play the game long enough, you’re going to have a lot of those days.’’

Bo Greenwell is a throwback. He’s cocky and has a swagger, but also possesses charisma and a love of the game. He wants to be a Hall of Famer and swears he’ll make it to The Show. He’s even picked the date: Sept. 8, 2013 — a late season call-up.

“I want to make a mark on this game,’’ he says.

Mike, who listens to every game on the Internet, is a believer.

“I think he’ll get there, I really do,’’ says Greenwell. “To me, Bo is just like me, except he’s a faster runner. Some people think he’s an overachiever, and there’s some truth to that. When you tell him he can’t do something, he’s going to prove you wrong. Those are traits that have to be in you. You can’t teach that.’’

Greenwell played 12 seasons with the Red Sox as the left fielder, succeeding three Hall of Famers: Ted Williams, Carl Yastrzemski, and Jim Rice. He never quite achieved their lofty status, but he was a two-time All-Star with a.303 career batting average, 130 home runs, and 726 RBIs. He got his nickname after he captured an alligator in spring training in Winter Haven, Fla., taped its mouth shut, and put it in Ellis Burks’s locker. Playing in the midst of the Steroid Era, he was runner-up to Oakland’s Jose Canseco for the American League MVP award in 1988. After Canseco admitted his steroid use in his 2005 book “Juiced’’, the Gator demanded that he be named MVP.

Watching Son of Gator play is like stepping into a time machine and dialing in 1987, the year Greenwell was fourth in Rookie of the Year voting.

“Yes, sir, I’m honored to be compared to my dad,’’ Bo says politely. “If my dad had won that MVP and the rookie of the year, he would be in the Hall of Fame. I would never do steroids. I’d rather get another job.’’

Tonight, the first-place Captains are playing the Dayton Dragons.

“It’s scary,’’ says Dayton manager Todd Benzinger, who played with Greenwell on the Red Sox. “He’s a lot like his old man with the mannerisms, the way he walks to the plate and swings the bat. Plus, he can hit like him, too.’’

‘He’s got some talent’

There have been more than 100 father-son duos in the major leagues, according to Baseball Almanac. Usually, the son never lives up to the legacy of the father. But there are notable exceptions.

Ken Griffey Sr., now the Dayton first base coach, was a three-time All-Star, but his son, Ken Griffey Jr., is a surefire Hall of Famer. Together they made baseball history when they hit back-to-back home runs for the Seattle Mariners on Sept. 14, 1990.

He says the biggest hurdle for young players is the mental aspect of the game.

But he scoffs at Bo Greenwell’s “bull’s-eye’’ analogy.

“Greenwell doesn’t have to worry about pressure,’’ says Griffey Sr. “If he goes out here and plays the game the way he loves to — and you can tell he loves playing this game — he’ll be fine. There won’t be no pressure.’’

Drafted in the sixth round by the Indians in 2007, Greenwell hit just .215 for the Indians rookie team in the Gulf Coast League. He improved to .263 in 2008, and last year with Lake County, he hit .290.

“He’s been unbelievable,’’ says Lake County manager Ted Kubiak. “He’s got some pop in his bat, but the thing that impresses me is his attitude, his confidence, his self-esteem. To the point that maybe it gets overboard a little bit. ’’

Kubiak tells his young players they can come to him for advice, but they also can go to Bo, who is wise beyond his years.

Cleveland Indians director of player development Ross Atkins says he loves Greenwell’s “grit and perseverance. He puts bat on ball and has good speed. If he continues to do this, he won’t be there long.’’

But it could be a long, slow climb for the 6-foot, 200-pound Floridian.

“He’s limited by his position,’’ said one veteran baseball observer who requested anonymity. “He doesn’t have a strong enough arm to play center or right and as he ages and gets stronger, he loses speed.’’

Professional development

Greenwell already has played in front of the Green Monster in full uniform during a Family Day event in 1992, when he was 3 years old. A slim Roger Clemens offered him a ball in front of the Wall. The pictures are framed and up on the Greenwell mantelpiece.

Because he was only 9 when his father retired, his Red Sox memories are limited. Greenwell never will forget slurping ice cream in Fenway Park and hearing the horrifying pop of a Randy Johnson fastball hitting his father’s batting helmet.

“At first everyone was scared but dad got up and walked away. I went to so many games in so many parks. Most of the time we’d sit in stands but we’d get antsy, and mom would take us to the children’s day care place and we would stay there and play.’’

“I saw Wade Boggs a couple of times. He always kept to himself. He was a special character but when he spoke, you listened. I remember going in and giving Mo Vaughn a fist pound and saying good job. I saw Mo at the Red Sox Hall of Fame induction [2008]. He’s like, ‘Man, you’ve grown up, you look just like your dad.’ ’’

The young Greenwell says being around all the clubhouses kept him from being awestruck of baseball stars.

“You know what, I was more happy getting that cherry Coke in that clubhouse than to meet Mo Vaughn, or talk to Roger Clemens, or all the greats that were there.’’

In fact, at one point, Greenwell preferred football to baseball.

At Riverdale High School in Fort Myers he was a star quarterback who was on the road to a Division 1 scholarship before ripping an ACL in a spring football scrimmage.

“I was 17 years old and scared to death,’’ he remembers. “Sports was all I’ve known.’’

He stayed in that game, vowed to be back as a senior, and then threw a 42-yard touchdown pass on his first play from scrimmage.

He also hit .510 his senior year, but Greenwell says the Red Sox never looked at him.

“Did my dad lobby for me? Not so much,’’ he says. “Did he put the bug in their ear? Absolutely.’’

Greenwell teased Red Sox scout and former teammate Al Nipper about the injustice. The Yankees, meanwhile, called Greenwell in for a workout at Legends Field.

“They asked me what it’s like to be Mike Greenwell’s son,’’ he remembers. “I just gestured. There he was behind home plate, like 16 rows up, just hanging out. He had his flip-flops on the row in front of him, reading a newspaper, you couldn’t even see his face.

“That’s why I love him. He’s there for me when I need him, but is he not riding on every single swing — absolutely not. He’s allowed me to evolve by myself. If I take a fastball down the middle, you don’t see him throw his hands up in the air.’’

Dream scenario

This year, Mike Greenwell has stopped racing stock cars and trucks professionally, and juggled chores on his fruit and vegetable farm in order to help coach his 18-year-old-son, Garrett. The first baseman is bigger and has more power than his older brother.

“We’d be great teammates, but I’m better, ’’ says Bo, flashing a grin.

Greenwell thinks both sons will make it to the major leagues.

“The apple didn’t fall too far from the tree for my two boys,’’ says the proud father. “They have a huge upside. I tell the scouts these are some of the safest draft picks you’ll ever get because they have the work ethic and desire.’’

He says he dreams Bo plays for the Red Sox.

“It would be a wonderful thing. He’ll have an advantage. I’ll talk to him about the Wall. Nobody ever talked to me about the Wall till I got there, so I looked like an idiot.’’

Bo has the same dream.

“If I could pick anything that could happen in my life, that would be it, whether in a Red Sox uniform or not. Just to trot out at Fenway on the same grass that my dad played on, in front of the same wall that my dad played, that would be so surreal. All I can do is put a smile on my face and bust my butt every day.